cp's OEIS Frontend

A369676 a(n) = Product_{k=0..n} (5^k + 5^(n-k)).

%I A369676 #16 Feb 07 2024 07:49:49
%S A369676 2,36,6760,14288400,331135220000,87265295649000000,
%T A369676 252668462115852250000000,8322480168806663555062500000000,
%U A369676 3012058207750727786980181328125000000000,12401474551899042876552569922821191406250000000000,561039675887726306551826113078284190093383789062500000000000
%N A369676 a(n) = Product_{k=0..n} (5^k + 5^(n-k)).
%C A369676 From _Vaclav Kotesovec_, Feb 07 2024: (Start)
%C A369676 For q > 1, Product_{k=0..n} (q^k + q^(n-k)) ~ c * q^(3*n^2/4 + n), where
%C A369676 c = QPochhammer(-1, 1/q^2)^2/2 if n is even and
%C A369676 c = q^(1/4) * QPochhammer(-q, 1/q^2)^2 / (q + 1)^2 if n is odd. (End)
%F A369676 a(n) = Product_{k=0..n} (5^k + 5^(n-k)).
%F A369676 a(n) = 5^(n*(n+1)) * Product_{k=0..n} (1/5^k + 1/5^(n-k)).
%F A369676 a(n) = 5^(n*(n+1)/2) * Product_{k=0..n} (1 + 1/5^(n-2*k)).
%F A369676 From _Vaclav Kotesovec_, Feb 07 2024: (Start)
%F A369676 a(n) ~ c * 5^(3*n^2/4 + n), where
%F A369676 c = 2.170417138549358... = QPochhammer(-1, 1/25)^2/2 if n is even and
%F A369676 c = 2.189351749288445... = 5^(1/4) * QPochhammer(-5, 1/25)^2 / 36 if n is odd. (End)
%e A369676 a(0) = (1 + 1) = 2;
%e A369676 a(1) = (1 + 5)*(5 + 1) = 36;
%e A369676 a(2) = (1 + 5^2)*(5 + 5)*(5^2 + 1) = 6760;
%e A369676 a(3) = (1 + 5^3)*(5 + 5^2)*(5^2 + 5)*(5^3 + 1) = 14288400;
%e A369676 a(4) = (1 + 5^4)*(5 + 5^3)*(5^2 + 5^2)*(5^3 + 5)*(5^4 + 1) = 331135220000;
%e A369676 a(5) = (1 + 5^5)*(5 + 5^4)*(5^2 + 5^3)*(5^3 + 5^2)*(5^4 + 5)*(5^5 + 1) = 87265295649000000;
%e A369676 ...
%e A369676 RELATED SERIES.
%e A369676 Let F(x) be the g.f. of A369557, then
%e A369676 F(1/5) = 2 + 36/5^2 + 6760/5^6 + 14288400/5^12 + 331135220000/5^20 + 87265295649000000/5^30 + ... + a(n)/5^(n*(n+1)) + ... = 3.934732308501055907377639201049737298238369356...
%o A369676 (PARI) {a(n) = prod(k=0, n, 5^k + 5^(n-k))}
%o A369676 for(n=0, 15, print1(a(n), ", "))
%Y A369676 Cf. A369673, A369674, A369675, A369557.
%K A369676 nonn
%O A369676 0,1
%A A369676 _Paul D. Hanna_, Feb 06 2024